Abstract
: This lecture discusses recent interactions between convex
optimization and algebraic geometry. Given a semidefinite programming
problem which is specified by matrices over the rational numbers,
each coordinate of its optimal solution is an algebraic number. We
determine the degree of the minimal polynomials of these algebraic
numbers. This degree measures the intrinsic algebraic complexity of
solving semidefinite programs. Geometrically, it counts the critical
points of a linear function over all matrices of fixed rank in a
linear space of symmetric matrices.
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